Bayesian inference and prediction for mean-mixtures of normal distributions

TL;DR

This paper analyzes Bayesian predictive densities for mean-mixtures of multivariate normal distributions, demonstrating improvements over traditional methods under Kullback-Leibler loss, especially in higher dimensions.

Contribution

It provides explicit Bayesian density representations and shows how to improve upon the minimum risk equivariant density for dimensions four and above.

Findings

Bayesian densities outperform MRE density under KL loss in dimensions ≥4.

Explicit formulas for Bayesian posterior and predictive densities are derived.

Numerical evaluations support the theoretical improvements.

Abstract

We study frequentist risk properties of predictive density estimators for mean mixtures of multivariate normal distributions, involving an unknown location parameter $\theta \in \mathbb{R}^d$, and which include multivariate skew normal distributions. We provide explicit representations for Bayesian posterior and predictive densities, including the benchmark minimum risk equivariant (MRE) density, which is minimax and generalized Bayes with respect to an improper uniform density for $\theta$. For four dimensions or more, we obtain Bayesian densities that improve uniformly on the MRE density under Kullback-Leibler loss. We also provide plug-in type improvements, investigate implications for certain type of parametric restrictions on $\theta$, and illustrate and comment the findings based on numerical evaluations.